MATH 151, FALL 2005
COMMON EXAM III - VERSION A
NAME (print):
INSTRUCTOR:
SECTION NUMBER:
UIN:
DIRECTIONS:
1. The use of a calculator, laptop or computer is prohibited.
2. In Part 1 (Problems 1-13), mark the correct choice on your ScanTron form No. 815-E using a No. 2 pencil. For
your own records, also record your choices on your exam! ScanTrons will be collected from all examinees after 90
minutes and will not be returned.
3. In Part 2 (Problems 14-18), present your solutions in the space provided. Show all your work neatly and concisely
and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality
and correctness of the work leading up to it.
4. Be sure to write your name, section number and version letter of the exam on the ScanTron form.
THE AGGIE CODE OF HONOR
“An Aggie does not lie, cheat or steal, or tolerate those who do.”
Signature:
DO NOT WRITE BELOW!
Question
Points Awarded
Points
1-13
52
14
14
15
8
16
10
17
6
18
10
100
1
PART I
√
1. (4 pts) sin−1 ( 3/2) =
(a) π
π
(b)
3
π
(c)
6
π
(d)
8
(e) 0
2. (4 pts) Which of the following is an anti-derivative of the function g(x) = 4x3 + xe(x
(a) 12x2 + (2x2 + 1)e(x
(b) 12x2 + e(x
(c) x4 +
e(x
2
(d) x4 + e(x
2
2
2
2
)
)
)
)
(e) x4 + 2e(x
2
)
Exam continues on next page
2
2
)
?
3. (4 pts) If y = xx , then
dy
=
dx
(a) xx
(b) 1 + ln(x)
(c) xx ln(x)
(d) xx−1 (1 + ln(x))
(e) xx (1 + ln(x))
ln(1 − x) + x +
4. (4 pts) lim
x→0
x3
x2
2 =
(a) −3
1
(b) −
3
1
(c)
3
(d) 1
(e) 3
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3
5. (4 pts) The acceleration of a particle is given by a(t) = h1, 2ti . The initial velocity of the particle is v(0) = h0, 1i
and its initial position is r(0) = h−1, 1i . Find the position r(t) of the particle at time t = 1 .
(a) h1 , −1i
(b) h1/2 , 7/3i
(c) h−1/2 , 7/3i
(d) h1/2 , 3i
(e) h−1/2 , 3i
6. (4 pts) If g(x) = sin−1 (x) , then g 00 (1/2) =
8
25
16
−
25
0
√
4 3
9
√
2 3
9
(a) −
(b)
(c)
(d)
(e)
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4
7. (4 pts) The inflection points of g(x) = 23 (2x6 − 5x4 ) occur at
(a) only x = −1 and x = 1
(b) x = −1 , x = 0 and x = 1
p
p
(c) only x = − 5/3 and x = 5/3
p
p
(d) x = − 5/3 , x = 0 and x = 5/3
(e) only x = 0 and x = 1
8. (4 pts) If f is a differentiable function such that f (1) = 5 and f (3) = 8 , then the Mean Value Theorem implies
that there is a number c with 1 < c < 3 such that
(a) f (c) = 3/2
(b) f 0 (c) = 3/2
(c) f (c) = 6
(d) f 0 (c) = 6
(e) f 0 (c) = 0
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5
Consider the function f (x) = x3 − 12x + 5 in both problem #9 and problem #10.
9. (4 pts) Find the absolute minimum value of f (x) on the interval [−5, 3] .
(a) −60
(b) −11
(c) −4
(d) 21
(e) 42
10. (4 pts) Find the absolute maximum value of f (x) on the interval [−5, 3] .
(a) −60
(b) −11
(c) −4
(d) 21
(e) 42
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6
11. (4 pts) Which formula gives the area under the curve y = 1 + x3 above the x-axis between x = 2 and x = 6 ?
"
3 #
n
4X
4i
1+
(a) lim
n→∞ n
n
i=1
"
#
3
n
4X
4i
1+
(b)
n i=1
n
"
3 #
n
4i
4X
1+ 2+
(c) lim
n→∞ n
n
i=1
"
#
n
3
4i
4X
1+ 2+
(d)
n i=1
n
"
3 #
n
i
1X
1+ 2+
(e) lim
n→∞ n
n
i=1
12. (4 pts) A bacteria culture starts with 1000 bacteria and the growth rate is proportional to the number of bacteria.
After 2 hours the population is 9000. Find the number of bacteria after 3 hours.
(a) 18, 000
(b) 27, 000
(c) 36, 000
(d) 54, 000
(e) 81, 000
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7
13. (4 pts) The function f (x) has the following properties:
(i) f 0 (x) > 0 for x ∈ (1, 3)
(ii) f 0 (x) < 0 for x ∈ (0, 1) ∪ (3, 4)
(iii) f 00 (x) > 0 for x ∈ (0, 2)
(iv) f 00 (x) < 0 for x ∈ (2, 4)
Which of the following could be the graph of y = f (x) ?
0
1
2
3
4
0
(a)
0
1
2
(c)
3
4
1
2
3
0
4
x
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8
2
3
4
3
4
x
(d)
x
0
(e)
1
(b)
1
2
x
PART II
dy
for the following functions. Don0 t simplify!
dx
√
(a) (5 pts) y = ln(ln(1 + x ))
14. Find
√
(b) (5 pts) y = tan−1 ( x5 + 4 )
(c) (4 pts) y = (x + x2 )tan(x)
Exam continues on next page
9
15. (8 pts) A car’s brakes are applied when the car is moving at 88 ft/sec. The brakes provide a constant deceleration
of 40 ft/sec2 . How far does the car travel before coming to a stop?
Exam continues on next page
10
16. (10 pts) An open-topped rectangular box is to have a volume of 36000 cm3 . If its bottom is a rectangle whose
length is twice its width, what dimensions would minimize the total area of the box?
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11
2
17. (6 pts) Find lim (cos(2x))1/x .
x→0
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12
18. Consider the function f (x) = x3 − 3x2 + 3 .
(a) (3 pts) Find all critical points of f (x) .
(b) (4 pts) Classify the critical points of f (x) . Clearly justify your answer!.
(c) (3 pts) Find the point(s) of inflection of f (x) . Clearly justify your answer!.
End of exam
13